Chapter: Waves

• 1. Classification of Waves

  • Based on Medium Requirement
    • Mechanical Waves: Require material medium (e.g., sound waves, string waves)
    • Electromagnetic Waves: Do not require medium (e.g., light, X-rays)
    • Matter Waves: Associated with moving microscopic particles (e.g., electrons)
  • Based on Particle Vibration
    • Transverse Waves
      • Particles vibrate perpendicular to wave propagation direction
      • Travel in form of crests and troughs
      • Can be polarized (e.g., waves on a string, light)
    • Longitudinal Waves
      • Particles vibrate parallel to wave propagation direction
      • Travel in form of compressions and rarefactions
      • Cannot be polarized (e.g., sound waves in air)

• 2. Wave Parameters & Equation

  • Progressive Wave Equation: $y(x, t) = A \sin(kx - \omega t + \phi)$
  • Amplitude ($A$): Maximum displacement of particles from mean position
  • Wavelength ($\lambda$): Distance between two consecutive crests or compressions
  • Time Period ($T$): Time taken to complete one full oscillation
  • Frequency ($f$): Number of oscillations per second ($f = \frac{1}{T}$)
  • Angular Frequency ($\omega$): $\omega = 2\pi f = \frac{2\pi}{T}$
  • Wave Number / Propagation Constant ($k$): $k = \frac{2\pi}{\lambda}$
  • Wave Velocity ($v$): $v = f\lambda = \frac{\omega}{k}$
  • Phase ($\phi$): Describes state of motion at a given instant

• 3. Speed of Travelling Waves

  • Transverse Wave on Stretched String
    • Formula: $v = \sqrt{\frac{T}{\mu}}$
    • $T$: Tension in string
    • $\mu$: Linear mass density (mass per unit length)
  • Longitudinal Wave in Medium
    • Fluids: $v = \sqrt{\frac{B}{\rho}}$ ($B$ = Bulk Modulus, $\rho$ = Density)
    • Solids: $v = \sqrt{\frac{Y}{\rho}}$ ($Y$ = Young's Modulus)
  • Speed of Sound in Air
    • Newton's Formula: Assumed isothermal process ($v = \sqrt{\frac{P}{\rho}}$) - Gave incorrect value
    • Laplace Correction: Assumed adiabatic process ($v = \sqrt{\frac{\gamma P}{\rho}}$) - Correct value
    • Factors Affecting Speed: Increases with temperature and humidity; independent of pressure

• 4. Superposition Principle & Reflection

  • Principle of Superposition: Net displacement $y = y_1 + y_2 + \dots + y_n$
  • Reflection at Rigid Boundary
    • Wave reverses direction with $\pi$ phase change (180°)
  • Reflection at Open / Free Boundary
    • Wave reverses direction with zero phase change

• 5. Standing (Stationary) Waves

  • Formation: Superposition of two identical waves traveling in opposite directions
  • Characteristics
    • No net energy transfer
    • Nodes: Points of zero amplitude / maximum pressure variation
    • Antinodes: Points of maximum amplitude / minimum pressure variation
    • Distance between two consecutive nodes or antinodes = $\frac{\lambda}{2}$
    • Distance between a node and adjacent antinode = $\frac{\lambda}{4}$
  • Modes of Vibration
    • Stretched String (Fixed at both ends)
      • Fundamental frequency: $f_1 = \frac{v}{2L}$
      • Frequencies: $f_n = n f_1$ (Contains all harmonics)
    • Closed Organ Pipe (One end closed)
      • Fundamental frequency: $f_1 = \frac{v}{4L}$
      • Frequencies: $f_n = (2n-1) f_1$ (Contains odd harmonics only)
    • Open Organ Pipe (Both ends open)
      • Fundamental frequency: $f_1 = \frac{v}{2L}$
      • Frequencies: $f_n = n f_1$ (Contains all harmonics)

• 6. Beats

  • Definition: Periodic variations in sound intensity due to superposition of waves with slightly different frequencies
  • Beat Frequency: $f_{\text{beat}} = |f_1 - f_2|$
  • Condition: Frequency difference must be less than 10 Hz for human ear detection
  • Applications: Tuning musical instruments, detecting toxic gases in mines

• 7. Doppler Effect

  • Definition: Apparent change in frequency of a wave due to relative motion between source and observer
  • General Formula: $f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right)$
    • $v$: Speed of wave in medium
    • $v_o$: Speed of observer
    • $v_s$: Speed of source
  • Apparent Frequency Increases: When source and observer move towards each other
  • Apparent Frequency Decreases: When source and observer move away from each other